# Show that sum of the two closed sets in $\mathbb{R^2}$ is not closed.

Show that sum of the two closed sets in $$\mathbb{R^2}$$ is not always closed.

In $$\mathbb{R^2}$$, consider the sets $$A =\{(x,y)\,\,:\,\, x>0,\,\,xy=1\}$$ and $$B =\{(x,y)\,\,:\,\, x>0,\,\,xy=-1\}$$ which are closed, show that the sum A + B is not closed. $$A+B:=\{a+b : a \in A, b \in B\}$$

To have better intuition, you can visualize the two sets as the following picture

• ${}$Nice example! – Lord Shark the Unknown Oct 10 '18 at 18:45
• What's the question? – Yanko Oct 10 '18 at 18:56
• I revised the question. Show that the set $A+B$ is not closed. – Saeed Oct 10 '18 at 19:03
• What subset of the plane is $A+B$? – rogerl Oct 10 '18 at 19:56
• What do you mean? – Saeed Oct 10 '18 at 23:11

Consider the sequence $$(x_n)=\Big\{\Big(\frac{2}{n},0\Big)\Big\}_{1}^\infty=\Big\{\Big(\frac{1}{n},n\Big)+\Big(\frac{1}{n},-n\Big)\Big\}_{1}^\infty \in A+B$$ Here $$x_n \longrightarrow (0,0) \notin A+B$$ since the first coordinate of the element in $$A+B$$ is always positive .
So, $$A+B$$ is not closed!